Solving X In The Equation X² - 4x - 9 = 29 A Step-by-Step Guide
In this article, we will delve into the step-by-step process of solving for x in the quadratic equation x² - 4x - 9 = 29. Quadratic equations, characterized by the presence of a squared variable, often require specific techniques to find their solutions. We will explore the method of completing the square, a powerful algebraic tool that allows us to rewrite the equation into a form where x can be easily isolated. This method not only provides the solutions but also enhances our understanding of the structure of quadratic equations. Understanding how to solve quadratic equations is a fundamental skill in algebra, with applications spanning various fields including physics, engineering, and economics. This article aims to provide a comprehensive guide to solving the given equation, ensuring clarity and precision in each step. Whether you are a student learning algebra or someone seeking to refresh your mathematical skills, this detailed solution will help you master the technique of completing the square and confidently solve similar problems.
Before we dive into the solution, it's crucial to understand what a quadratic equation is and the general approaches to solving them. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. These equations appear frequently in various mathematical and real-world contexts, such as physics (projectile motion), engineering (designing parabolic structures), and economics (modeling cost and revenue). The solutions to a quadratic equation, also known as roots or zeros, represent the values of x that satisfy the equation. There are several methods to find these solutions, each with its own advantages and applications. Factoring, for instance, is a straightforward method when the quadratic expression can be easily factored into two linear expressions. The quadratic formula is a universal method that can solve any quadratic equation, regardless of its factorability. Completing the square, the method we will use in this article, is another powerful technique that not only solves the equation but also provides insights into the structure and properties of quadratic expressions. Understanding the nature and behavior of quadratic equations is essential for anyone working with mathematical models and problem-solving in a variety of disciplines. Each method offers a unique perspective and contributes to a deeper understanding of algebraic concepts.
Completing the square is a technique used to rewrite a quadratic equation in the form (x - h)² = k, which makes it easier to solve for x. This method involves manipulating the equation algebraically to create a perfect square trinomial on one side. The process consists of several key steps, each carefully designed to transform the equation while preserving its equality. First, we ensure that the coefficient of the x² term is 1. If it isn't, we divide the entire equation by that coefficient. Next, we move the constant term to the right side of the equation. This sets the stage for creating the perfect square trinomial. The core of completing the square involves taking half of the coefficient of the x term, squaring it, and adding it to both sides of the equation. This step is crucial because it creates a perfect square trinomial on the left side, which can then be factored into the form (x - h)². The final steps involve taking the square root of both sides and solving for x. Completing the square is not just a method for solving quadratic equations; it is also a powerful tool for understanding the structure of quadratic expressions. It allows us to identify the vertex of a parabola (the graph of a quadratic function) and rewrite quadratic expressions in vertex form, which provides valuable information about the function's behavior. This technique is particularly useful when factoring is difficult or impossible, and it offers a deeper understanding of algebraic manipulation and problem-solving strategies.
Step-by-Step Solution
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Rewrite the Equation: First, we rewrite the given equation x² - 4x - 9 = 29 by adding 9 to both sides to isolate the terms with x on one side:
x² - 4x = 38
This step simplifies the equation and prepares it for the process of completing the square. By isolating the terms with x, we can focus on creating a perfect square trinomial. Adding the same value to both sides maintains the equation's balance and ensures that the solutions remain unchanged. This initial step is a fundamental aspect of solving algebraic equations, emphasizing the importance of isolating variables and simplifying expressions to make them easier to work with. The clarity of this step is essential for the subsequent steps, as any errors here can propagate through the rest of the solution. Careful execution of this step sets the stage for a successful application of the completing the square method.
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Complete the Square: To complete the square, we need to add a value to both sides of the equation that will make the left side a perfect square trinomial. We take half of the coefficient of the x term (-4), which is -2, and square it: (-2)² = 4. Adding 4 to both sides gives us:
x² - 4x + 4 = 38 + 4
x² - 4x + 4 = 42
The core of completing the square lies in this step. By adding the appropriate value, we transform the left side of the equation into a perfect square trinomial, which is crucial for simplifying and solving the equation. Taking half of the coefficient of the x term and squaring it is the key to determining the value that completes the square. This process ensures that the resulting trinomial can be factored into a binomial squared. Adding the same value to both sides maintains the equation's balance and ensures that the solutions remain valid. This step is a critical component of the completing the square method, and a thorough understanding of this process is essential for mastering this technique.
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Factor the Left Side: The left side of the equation is now a perfect square trinomial and can be factored as follows:
(x - 2)² = 42
This step simplifies the equation significantly by expressing the quadratic expression as a squared binomial. Recognizing and factoring the perfect square trinomial is a key skill in algebra, allowing for easier manipulation and solution of quadratic equations. The factored form directly reveals the value that, when subtracted from x, results in a perfect square. This transformation is a crucial part of the completing the square method, as it sets the stage for isolating x. The ability to factor perfect square trinomials accurately is essential for successful application of this technique, and it underscores the importance of understanding the relationship between factored forms and expanded expressions. This step efficiently condenses the equation into a more manageable form, paving the way for the final solution.
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Take the Square Root: Next, take the square root of both sides of the equation:
√(x - 2)² = ±√42
x - 2 = ±√42
This step introduces the concept of both positive and negative square roots, which is crucial for finding all possible solutions to the quadratic equation. Taking the square root of both sides is a direct application of algebraic principles, but it is essential to remember that the square root operation yields two possible results: a positive and a negative value. This duality is a fundamental aspect of quadratic equations and their solutions. The ± symbol indicates that we must consider both the positive and negative square roots to find all valid solutions for x. This step directly simplifies the equation and brings us closer to isolating x. A thorough understanding of square roots and their properties is essential for correctly executing this step and obtaining all possible solutions to the equation.
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Solve for x: Finally, add 2 to both sides to solve for x:
x = 2 ± √42
This final step isolates x and provides the solutions to the original quadratic equation. By adding 2 to both sides, we effectively undo the subtraction in the binomial, leaving x by itself. The solutions are expressed in the form 2 ± √42, indicating that there are two possible values for x: 2 + √42 and 2 - √42. This step is the culmination of the completing the square method, and it demonstrates the power of algebraic manipulation in solving equations. The solutions represent the values of x that satisfy the original quadratic equation, and they can be used for further analysis or application in various contexts. This final step underscores the importance of algebraic precision and the ability to correctly apply mathematical operations to solve for unknown variables.
Final Answer
Therefore, the solutions for x in the equation x² - 4x - 9 = 29 are:
x = 2 ± √42
This corresponds to option A.
In conclusion, we have successfully solved the quadratic equation x² - 4x - 9 = 29 using the method of completing the square. This method is a powerful tool for solving quadratic equations and provides valuable insights into the structure of quadratic expressions. The steps involved include rewriting the equation, completing the square, factoring the perfect square trinomial, taking the square root of both sides, and solving for x. By meticulously following these steps, we arrived at the solutions x = 2 ± √42. Understanding and mastering the technique of completing the square is essential for students and professionals alike, as quadratic equations appear in various mathematical and real-world contexts. This detailed solution demonstrates the importance of algebraic manipulation and problem-solving strategies in mathematics. The ability to solve quadratic equations effectively opens doors to further exploration in algebra, calculus, and other related fields. The method of completing the square not only provides the solutions but also enhances our understanding of the properties and behavior of quadratic expressions. Therefore, mastering this technique is a valuable investment in mathematical proficiency.
